On eigenmeasures under Fourier transform

TL;DR

This paper characterizes and classifies eigenmeasures of the Fourier transform on tempered measures, revealing connections with discrete Fourier transform, Meyer sets, and constructing new examples with specific support properties.

Contribution

It provides a comprehensive classification of eigenmeasures, including periodic, uniformly discrete, and novel support configurations, linking Fourier analysis with aperiodic order.

Findings

Classified all periodic eigenmeasures on \\RR.

Connected eigenmeasures with discrete support to Meyer sets.

Constructed eigenmeasures with non-uniform support and large gaps.

Abstract

Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on $\RR^d$. In particular, we classify all periodic eigenmeasures on $\RR$, which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on $\RR$ with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around $0$.